Dual Park-Ravani Interpolation of Rigid Motions: Acceleration-Field Continuity and Holonomic Hermite Repair
arXiv:2608.28826v1 Announce Type: new Abstract: The Park-Ravani construction generates a twice continuously differentiable, frame-invariant spline on SO(3) by exponentiating cubic canonical-coordinate polynomials. We show that the construction transfers, without changing form, to the group of orthogonal dual tensors, a representation of rigid displacements. The transferred recurrence is stated compactly through the dual extension of the right Jacobian of the exponential map and its first Fr\'echet derivative. This yields interpolation of prescribed rigid poses and continuity of the body dual twist and its first derivative. Using the higher-order rigid-body kinematics of dual spatial twists, we then prove that the resulting curve has a continuous physical acceleration field, not merely a continuous quantity obtained by formally differentiating the dual part of a twist. We also distinguish algebraic dual transfer from temporal differential prolongation: their simultaneous first-order use takes place in a hyper-dual algebra, and interpolation of arbitrary prolonged nodal data need not be holonomic. A noncommuting three-pose example verifies the recurrence, all knot continuity statements, and dimensional covariance under a change from meters to millimeters. We define and analyze the first-order holonomy defect of a generic hyper-dual interpolant, exhibit an exact counterexample, and remove the defect by cubic or quintic Hermite interpolation in dual logarithmic coordinates.
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[Submitted on 28 Aug 2026]
Title:Dual Park-Ravani Interpolation of Rigid Motions: Acceleration-Field Continuity and Holonomic Hermite Repair
View a PDF of the paper titled Dual Park-Ravani Interpolation of Rigid Motions: Acceleration-Field Continuity and Holonomic Hermite Repair, by Daniel Condurache
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Abstract:The Park-Ravani construction generates a twice continuously differentiable, frame-invariant spline on SO(3) by exponentiating cubic canonical-coordinate polynomials. We show that the construction transfers, without changing form, to the group of orthogonal dual tensors, a representation of rigid displacements. The transferred recurrence is stated compactly through the dual extension of the right Jacobian of the exponential map and its first Fréchet derivative. This yields interpolation of prescribed rigid poses and continuity of the body dual twist and its first derivative. Using the higher-order rigid-body kinematics of dual spatial twists, we then prove that the resulting curve has a continuous physical acceleration field, not merely a continuous quantity obtained by formally differentiating the dual part of a twist. We also distinguish algebraic dual transfer from temporal differential prolongation: their simultaneous first-order use takes place in a hyper-dual algebra, and interpolation of arbitrary prolonged nodal data need not be holonomic. A noncommuting three-pose example verifies the recurrence, all knot continuity statements, and dimensional covariance under a change from meters to millimeters. We define and analyze the first-order holonomy defect of a generic hyper-dual interpolant, exhibit an exact counterexample, and remove the defect by cubic or quintic Hermite interpolation in dual logarithmic coordinates.
Comments: 12 pages. Ancillary Python programs reproduce the numerical verification
Subjects:
Robotics (cs.RO)
MSC classes: 65D05, 65D17, 70B10, 22E70
Cite as: arXiv:2608.28826 [cs.RO]
(or arXiv:2608.28826v1 [cs.RO] for this version)
https://doi.org/10.48550/arXiv.2608.28826
arXiv-issued DOI via DataCite (pending registration)
Submission history
From: Daniel Condurache [view email] [v1] Fri, 28 Aug 2026 19:54:39 UTC (20 KB)
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